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Betaexpansions for cubic Pisot numbers
"... Real numbers can be represented in an arbitrary base > 1 using the transformation T : x ! x (mod 1) of the unit interval; any real number x 2 [0; 1] is then expanded into d (x) = (x i ) i1 where x i = b T (x)c. ..."
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Cited by 15 (0 self)
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Real numbers can be represented in an arbitrary base > 1 using the transformation T : x ! x (mod 1) of the unit interval; any real number x 2 [0; 1] is then expanded into d (x) = (x i ) i1 where x i = b T (x)c.
BETAEXPANSIONS OF RATIONAL NUMBERS IN QUADRATIC PISOT BASES
"... Abstract. We study rational numbers with purely periodic Rényi βexpansions. For bases β satisfying β2 = aβ + b with b dividing a, we give a necessary and sufficient condition for γ(β) = 1, i.e., that all rational numbers p/q ∈ [0, 1) with gcd(q, b) = 1 have a purely periodic βexpansion. A simple ..."
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Abstract. We study rational numbers with purely periodic Rényi βexpansions. For bases β satisfying β2 = aβ + b with b dividing a, we give a necessary and sufficient condition for γ(β) = 1, i.e., that all rational numbers p/q ∈ [0, 1) with gcd(q, b) = 1 have a purely periodic βexpansion. A
BETAEXPANSIONS, NATURAL EXTENSIONS AND MULTIPLE TILINGS ASSOCIATED WITH PISOT UNITS
"... Abstract. From the works of Rauzy and Thurston, we know how to construct (multiple) tilings of some Euclidean space using the conjugates of a Pisot unit β and the greedy βtransformation. In this paper, we consider different transformations generating expansions in base β, including cases where the ..."
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Cited by 12 (7 self)
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Abstract. From the works of Rauzy and Thurston, we know how to construct (multiple) tilings of some Euclidean space using the conjugates of a Pisot unit β and the greedy βtransformation. In this paper, we consider different transformations generating expansions in base β, including cases where
Betaexpansions for infinite families of Pisot and Salem numbers
 J. Number Theory
"... Abstract. This paper continues the study of betaexpansions of 1 where β is a Pisot or Salem number. Special attention is given to regular Pisot numbers, and the Salem numbers that approach these Pisot numbers. 1. ..."
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Abstract. This paper continues the study of betaexpansions of 1 where β is a Pisot or Salem number. Special attention is given to regular Pisot numbers, and the Salem numbers that approach these Pisot numbers. 1.
betaEXPANSIONS WITH DELETED DIGITS FOR PISOT NUMBERS beta
, 1995
"... An algorithm is given for computing the Hausdorff dimension of the set(s) = (fi ; D) of real numbers with representations x = P 1 n=1 dn fi \Gamman , where each dn 2 D, a finite set of "digits", and fi ? 0 is a Pisot number. The Hausdorff dimension is shown to be log = log fi, where ..."
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An algorithm is given for computing the Hausdorff dimension of the set(s) = (fi ; D) of real numbers with representations x = P 1 n=1 dn fi \Gamman , where each dn 2 D, a finite set of "digits", and fi ? 0 is a Pisot number. The Hausdorff dimension is shown to be log = log fi, where
Cubic Pisot units with finite beta expansions
 In Algebraic number theory and Diophantine analysis
, 1998
"... Abstract. Cubic Pisot units with finite beta expansion property are classified (Theorem 3). The results of [6] and [3] are well combined to complete its proof. Further, it is noted that the above finiteness property is equivalent to an important problem of fractal tiling generated by Pisot numbers. ..."
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Cited by 45 (5 self)
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Abstract. Cubic Pisot units with finite beta expansion property are classified (Theorem 3). The results of [6] and [3] are well combined to complete its proof. Further, it is noted that the above finiteness property is equivalent to an important problem of fractal tiling generated by Pisot numbers
Arithmetics on betaexpansions
, 2001
"... In this paper we consider representation of numbers in an irrational basis β> 1. We study the arithmetic operations on βexpansions and provide bounds on the number of fractional digits arising in addition and multiplication, L⊕(β) and L(β), respectively. We determine these bounds for irrational ..."
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for quadratic Pisot units to other quadratic Pisot numbers. 1 Betaexpansions Let β be a real number strictly greater than 1. A real number x ≥ 0 can be represented using a sequence (xi)k≥i>−∞, xi ∈ Z, 0 ≤ xi < β, such that
GROWTH RATE FOR BETAEXPANSIONS
"... Abstract. Let β> 1 and let m> β be an integer. Each x ∈ Iβ: = [0, m−1 β−1] can be represented in the form x = εkβ −k, k=1 where εk ∈ {0, 1,..., m − 1} for all k (a βexpansion of x). It is known that a.e. x ∈ Iβ has a continuum of distinct βexpansions. In this paper we prove that if β is a Pi ..."
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Pisot number, then for a.e. x this continuum has one and the same growth rate. We also link this rate to the Lebesguegeneric local dimension for the Bernoulli convolution parametrized by β. When β < 1+√5 2, we show that the set of βexpansions grows exponentially for every internal x. 1.
Results 1  10
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11,905